Recovery of tunable bound states in the continuum
Tunable bound states in the continuum (BICs) in photonic crystal slabs are highly sensitive to substrate-induced mirror-symmetry breaking and typically degrade into finite-$Q$ quasi-BICs in realistic integrated platforms. Here we show that such degradation can be deterministically reversed. Using temporal coupled-mode theory and full-wave simulations, we demonstrate that the radiation channel opened by the substrate can be exactly cancelled by introducing a second, independent odd-parity perturbation inside the slab. This dual-asymmetry strategy restores the singularity of the radiation matrix and thereby recovers a tunable BIC in a substrate-supported photonic crystal slab. The recovered state regains both the polarization vortex and the characteristic $Q\propto \Delta k^{-2}$ scaling. The recovery points further follow a linear relation in the two-asymmetry parameter space, revealing a simple mode-dependent compensation law. The same mechanism also restores merging-BIC configurations, showing that it applies not only to isolated tunable BICs but also to higher-order topological resonance states built from them. Our results establish a practical route for preserving tunable topological resonances in substrate-supported nanophotonic systems.